Linear-Algebra

## Covectors

A linear mapping from a vector space to a field of scalars. In other words, a linear function which acts upon a vector resulting in a real number (scalar) \begin{equation} \alpha\,:\,\mathbf{V} \longrightarrow \mathbb{R} \end{equation} Simplistically, covectors can be thought of as “row vectors”, or: \begin{equation} \begin{bmatrix} 1 & 2 \end{bmatrix} \end{equation} This might look like a standard vector, which would be true in an orthonormal basis, but it is not true generally. ...

## Differential Geometry

Tensors Tensor Product

## Linear Mapping

A mapping from $$\mathbf{V} \rightarrow \mathbf{W}$$ that preserves the operations of addition and scalar multiplication. Also known as # Linear Map Linear Transformation Linear Function

## Multilinear Map

A function of several variables that is linear, separately, in each variable. A multilinear map of one variable is a standard linear mapping.

## Dual Space

The space of all linear functionals $$f:V\rightarrow \mathbb{R}$$, noted as $$V^{*}$$ The dual space has the same dimension as the corresponding vector space or, given a space $$V$$, with bases $$(v_{1},…,v_{n})$$, there exists a dual space $$V^{*}$$ with a dual basis $$(v^{*}_{1},…,v^{*}_{n})$$.

## Dual Vector Space

The space of all linear functionals $$f:V\rightarrow \mathbb{R}$$, noted as $$V^{*}$$ The dual space has the same dimension as the corresponding vector space or, given a space $$V$$, with bases $$(v_{1},…,v_{n})$$, there exists a dual space $$V^{*}$$ with a dual basis $$(v^{*}_{1},…,v^{*}_{n})$$.

## Tensors

As a linear representation # A tensor can be represented as a vector of x-number of dimensions. Basically, a generalization on top of scalars, vectors, and matrices. The specific “flavor” of the tensor (i.e. is it a scalar, vector, or matrix) is clarified by referring to the tensor’s “rank”. For instance; a rank 0 tensor is a scalr, rank 1 tensor is a one-dimensional vector, a rank 2 tensor is a two-dimensional vector ($$2x2$$ matrix), etc. ...

## Bases

a basis for an n-dimensional vector space $$V$$ is any ordered set of linearly independent vectors $$(\mathbf{e}_{1}, \mathbf{e}_{2},…,\mathbf{e}_{n})$$ An arbitrary vector $$\mathbf{x}$$ in $$V$$ can be expressed as a linear combination of the basis vectors: \begin{equation} \mathbf{x}\,=\,\sum\limits_{i = 1}^{n} \mathbf{e}_{i}x^{i} \end{equation} See Bases Transformation, Coordinate Transformation

## Orthonormal Basis

An orthonormal basis is a basis where all the vectors are one unit long and all perpendicular to each other (e.g. the Cartesian plane)